Real-rooted atlas

A136572 → basis A013610 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(1 - 3\,x + 9\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(n + 9\,n x + 27\,n x^{2} + 27\,n x^{3} + 9\,n^{2} x + 45\,n^{2} x^{2} + 54\,n^{2} x^{3} + 9\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111213
2248162212
36216145825924272
4241152116644147248600102912
5120720097200518400121500010497602887680
672051840874800622080021870000377913602571912092528640
7504042336085730407620480035721000092588832012602368807054387203333980160

All 15 rows

Technical details
Node id
n-b8fc3e74039cf5b1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-10c18c8945c6e24a

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      1,       12;
  2:      2,       48,        162;
  3:      6,      216,       1458,        2592;
  4:     24,     1152,      11664,       41472,       48600;
  5:    120,     7200,      97200,      518400,     1215000,      1049760;
  6:    720,    51840,     874800,     6220800,    21870000,     37791360,      25719120;
  7:   5040,   423360,    8573040,    76204800,   357210000,    925888320,    1260236880,     705438720;
  8:  40320,  3870720,   91445760,   975421440,  5715360000,  19752284160,   40327580160,   45148078080,   21427701120;
  9: 362880, 39191040, 1058158080, 13168189440, 92588832000, 399983754240, 1088844664320, 1828497162240, 1735643790720, 714256704000;
%F Formula
The row polynomials satisfy (1+(9*n-3)*x+9*n^2*x^2)*P(n,x) = (n+(9*n^2+9*n)*x+(9*n^3+45*n^2+27*n)*x^2+(27*n^3+54*n^2+27*n)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (9*n^2 + 9*n)*x + (9*n^3 + 45*n^2 + 27*n)*x^2 + (27*n^3 + 54*n^2 + 27*n)*x^3)*P[n - 1, x])/(1 + (9*n - 3)*x + 9*n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + (9*n^2 + 9*n)*x + (9*n^3 + 45*n^2 + 27*n)*x^2 + (27*n^3 + 54*n^2 + 27*n)*x^3)*v[i - 1])/(1 + (9*n - 3)*x + 9*n^2*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.